@article{19628, abstract = {We consider the critical temperature for superconductivity, defined via the linear BCS equation. We prove that at weak coupling the critical temperature for a sample confined to a quadrant in two dimensions is strictly larger than the one for a half-space, which in turn is strictly larger than the one for R^2. Furthermore, we prove that the relative difference of the critical temperatures vanishes in the weak coupling limit.}, author = {Roos, Barbara and Seiringer, Robert}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Enhanced superconductivity at a corner for the linear BCS equation}}, doi = {10.1017/fms.2024.145}, volume = {13}, year = {2025}, } @article{13178, abstract = {We consider the large polaron described by the Fröhlich Hamiltonian and study its energy-momentum relation defined as the lowest possible energy as a function of the total momentum. Using a suitable family of trial states, we derive an optimal parabolic upper bound for the energy-momentum relation in the limit of strong coupling. The upper bound consists of a momentum independent term that agrees with the predicted two-term expansion for the ground state energy of the strongly coupled polaron at rest and a term that is quadratic in the momentum with coefficient given by the inverse of twice the classical effective mass introduced by Landau and Pekar.}, author = {Mitrouskas, David Johannes and Mysliwy, Krzysztof and Seiringer, Robert}, issn = {2050-5094}, journal = {Forum of Mathematics}, pages = {1--52}, publisher = {Cambridge University Press}, title = {{Optimal parabolic upper bound for the energy-momentum relation of a strongly coupled polaron}}, doi = {10.1017/fms.2023.45}, volume = {11}, year = {2023}, } @article{14239, abstract = {Given a resolution of rational singularities π:X~→X over a field of characteristic zero, we use a Hodge-theoretic argument to prove that the image of the functor Rπ∗:Db(X~)→Db(X) between bounded derived categories of coherent sheaves generates Db(X) as a triangulated category. This gives a weak version of the Bondal–Orlov localization conjecture [BO02], answering a question from [PS21]. The same result is established more generally for proper (not necessarily birational) morphisms π:X~→X , with X~ smooth, satisfying Rπ∗(OX~)=OX .}, author = {Mauri, Mirko and Shinder, Evgeny}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Homological Bondal-Orlov localization conjecture for rational singularities}}, doi = {10.1017/fms.2023.65}, volume = {11}, year = {2023}, } @article{14343, abstract = {The total energy of an eigenstate in a composite quantum system tends to be distributed equally among its constituents. We identify the quantum fluctuation around this equipartition principle in the simplest disordered quantum system consisting of linear combinations of Wigner matrices. As our main ingredient, we prove the Eigenstate Thermalisation Hypothesis and Gaussian fluctuation for general quadratic forms of the bulk eigenvectors of Wigner matrices with an arbitrary deformation.}, author = {Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha and Kolupaiev, Oleksii}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Gaussian fluctuations in the equipartition principle for Wigner matrices}}, doi = {10.1017/fms.2023.70}, volume = {11}, year = {2023}, } @article{10643, abstract = {We prove a generalised super-adiabatic theorem for extended fermionic systems assuming a spectral gap only in the bulk. More precisely, we assume that the infinite system has a unique ground state and that the corresponding Gelfand–Naimark–Segal Hamiltonian has a spectral gap above its eigenvalue zero. Moreover, we show that a similar adiabatic theorem also holds in the bulk of finite systems up to errors that vanish faster than any inverse power of the system size, although the corresponding finite-volume Hamiltonians need not have a spectral gap. }, author = {Henheik, Sven Joscha and Teufel, Stefan}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, keywords = {computational mathematics, discrete mathematics and combinatorics, geometry and topology, mathematical physics, statistics and probability, algebra and number theory, theoretical computer science, analysis}, publisher = {Cambridge University Press}, title = {{Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk}}, doi = {10.1017/fms.2021.80}, volume = {10}, year = {2022}, } @article{9318, abstract = {We consider a system of N bosons in the mean-field scaling regime for a class of interactions including the repulsive Coulomb potential. We derive an asymptotic expansion of the low-energy eigenstates and the corresponding energies, which provides corrections to Bogoliubov theory to any order in 1/N.}, author = {Bossmann, Lea and Petrat, Sören P and Seiringer, Robert}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Asymptotic expansion of low-energy excitations for weakly interacting bosons}}, doi = {10.1017/fms.2021.22}, volume = {9}, year = {2021}, } @article{9550, abstract = {We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices. }, author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Equipartition principle for Wigner matrices}}, doi = {10.1017/fms.2021.38}, volume = {9}, year = {2021}, } @article{7790, abstract = {We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density 𝜌 and inverse temperature 𝛽 differs from the one of the noninteracting system by the correction term 𝜋𝜌𝜌𝛽𝛽 . Here, is the scattering length of the interaction potential, and 𝛽 is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. The result is valid in the dilute limit 𝜌 and if 𝛽𝜌 .}, author = {Deuchert, Andreas and Mayer, Simon and Seiringer, Robert}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{The free energy of the two-dimensional dilute Bose gas. I. Lower bound}}, doi = {10.1017/fms.2020.17}, volume = {8}, year = {2020}, } @article{9583, abstract = {We show that for any n divisible by 3, almost all order-n Steiner triple systems admit a decomposition of almost all their triples into disjoint perfect matchings (that is, almost all Steiner triple systems are almost resolvable).}, author = {Ferber, Asaf and Kwan, Matthew Alan}, issn = {2050-5094}, journal = {Forum of Mathematics}, publisher = {Cambridge University Press}, title = {{Almost all Steiner triple systems are almost resolvable}}, doi = {10.1017/fms.2020.29}, volume = {8}, year = {2020}, } @article{6182, abstract = {We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields 173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.}, author = {Erdös, László and Krüger, Torben H and Schröder, Dominik J}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Random matrices with slow correlation decay}}, doi = {10.1017/fms.2019.2}, volume = {7}, year = {2019}, }